M3 — Matrix-Free Measurement Mitigation

qc/mitigation qc/qiskit

A scalable readout-error mitigation method that corrects measurement bitstring counts in post-processing without ever building the full $2^n\times 2^n$ assignment matrix that naive readout correction requires. M3 assumes correlations of order three and higher between qubit readout errors are negligible, so it only needs small $2\times 2$ (single-qubit) and $4\times 4$ (two-qubit) assignment matrices, built by preparing all qubits in $|0\dots 0⟩$ and $|1\dots 1⟩$ and measuring their error rates. These small matrices are then combined only for the bitstrings actually observed in a run, so the effective mitigation matrix scales with the number of distinct sampled outcomes, not $2^n$ — which is what lets M3 mitigate circuits with far more qubits than a full linear-algebra inversion could handle. Key insight: M3 is a post-processing correction applied to counts you already have, unlike TREX, which changes what happens during circuit execution (randomized bit-flips before measurement) to make the readout-error matrix invertible in the first place — the two are complementary, not competing: TREX conditions the noise, M3 corrects the counts after the fact.

import mthree
 
meas_map = mthree.utils.final_measurement_mapping(optimized_circuit)
mit = mthree.M3Mitigation(backend)
mit.cals_from_system(meas_map)
 
quasi_probs = mit.apply_correction(counts, meas_map)  # quasi-probability distribution

API notes

• mthree.utils.final_measurement_mapping(circuit) recovers which physical qubits the classical bits actually correspond to after transpilation/routing — required before calibration, since M3 needs to calibrate the exact physical qubits used, not the virtual/logical ones.
• .cals_from_system(meas_map) runs the $|0\dots 0⟩$/$|1\dots 1⟩$ calibration circuits on the backend for just the qubits in meas_map.
• .apply_correction(counts, meas_map) returns quasi-probabilities (can include small negative values from the correction), not counts — convert back to a valid distribution by clipping negatives to zero and renormalizing before further analysis.
• M3 correction is applied on top of raw sampled counts, so it composes with circuit-level suppression techniques (dynamical decoupling, gate/measurement twirling) applied beforehand — mitigating readout error doesn’t preclude also suppressing coherent error during execution.

Related

• EstimatorOptions and the Five Mitigation Knobs — TREX is the Estimator-side readout mitigation; M3 is the Sampler-side, post-processing alternative
• PEC — Probabilistic Error Cancellation
• Zero Noise Extrapolation (ZNE)
• The Partition Problem — QAOA Worked Example — M3 is compared against no-mitigation, suppression-only, and combined configurations on this worked example

Self-Check

• Why does M3 avoid building the full $2^n\times 2^n$ assignment matrix, and what assumption makes that possible?
• What’s the difference between what TREX does and what M3 does, and why are they complementary rather than redundant?
• Why does apply_correction return quasi-probabilities that can include negative values, and what do you need to do before treating them as a normal distribution?